Bethe theory

Oddershede s earlier results [3-5] calculate the directional values of the dipole oscillator strength distribution for use in the Bethe theory [9], which is valid for high-energy projectiles. Our approach, since we have not implemented the possibility of treating unbound electrons, is restricted to calculating stopping cross... 

Being based on lowest-order perturbation theory, Bethe theory approximates equation (3) by T—2 Zie lp) lmv so that... 

A most remarkable feature of Bethe s theory, pointed out by Bloch [8], is that p ax is the same in Bohr and Bethe theory, not only in the present, simplified picture but also in the more detailed mathematical form. 

In Bethe theory the shell correction ALsheii is conveniently defined as the difference between the stopping number LBom in the Born approximation and the Bethe logarithm LBethe —in (2mv /I). Fano [12] wrote the leading correction in the form... 

It was a fairly common belief that such a theory should be a modification of the Bethe theory. Although the fact that the screening regime (uclassical theory must be more powerful than the Bom approximation in this regime was evidently not taken at face value. 

Bethe theory was reevaluated on the basis of screened instead of unscreened Coulomb interaction [42],... 

In the Bethe theory of stopping, the GOS characterize the excitation spectrum of a target for a given momentum transfer q and they are given by... 

Because the semiclassical theories can be used to calculate differential cross sections with relative ease for close collisions between the incident charged particle and the bound electron, and the Bethe theory provides a straightforward method to describe low-energy electrons ejected in distant collisions, it is only natural to combine the best characteristics of the two approaches to derive a comprehensive description of electron impact ionization. 

The relative success of the binary encounter and Bethe theories, and the relatively well established systematic trends observed in the measured differential cross sections for ionization by fast protons, has stimulated the development of models that can extend the range of data for use in various applications. It is clear that the low-energy portion of the secondary electron spectra are related to the optical oscillator strength and that the ejection of fast electrons can be predicted reasonable well by the binary encounter theory. The question is how to merge these two concepts to predict the full spectrum. 

According to the optical approximation, which was shown by Platzman [4] to be based on the Born-Bethe theory, a radiation chemical yield of channel G, may be estimated from optical data as shown in the following equation ... 

We will now summarize the conclusion of the Kirkwood-Bethe theory. Fig 15 shows the computed peak pressure and computed reduced tunc constant for TNT plotted VS the inverse reduced distance. The dotted lines are a power function fit thru the computed peak pressures. The x s are drawn in by the writer to compare computed and measured reduced time constants (taken from Fig 7.9, p 240 of Ref 1). Comparison of other computed and measured shock parameters on the basis of the power functions shown below (in Cole s notation and in English units) is made in Table 11 (from p 242 of Ref 1)... 

Values predicted from the Kirkwood-Bethe theory over the same pressure range are given in parentheses... 

Fig 15 Calculated Peak Pressure and Time Constant for TNT (Kirkwood-Bethe theory)... 

The pressures for spherical charges of the same weight as cylinders with length/radius ratios of 10 and 50 are plotted in Fig. 18, as calculated from the Kirkwood-Bethe theory. 

Introducing spin adds significant magnetic interactions, which may be accounted for with the Breit-Bethe theory.3 The theory is based on the unsymmetrized wavefunctions used to calculate the polarization energy. There are two terms which make significant contributions, the spin-orbit interaction and the spin-spin interactions. These operators have the matrix elements3... 

The binary-encounter-dipole (BED) model of Kim and Rudd [31] couples the modified form of Mott cross section [32] with the Bom-Bethe theory [27]. BED requires the differential continuum oscillator strength (DOS) which is rather difficult to obtain. The simplest approximate version of BED is the binary-encounter-Bethe (BEB) [31] model, which does not need the knowledge of DOS for calculating the EISICS. 

M. Inokuti, Inelastic collisions of fast charged particles with atoms and molecules - the Bethe theory revisited, Rev. Mod. Phys. 43 (1971) 297. 

Despite the kinship of the present development to these historical works, we don t try to follow upon the historical footsteps directly. Part of our reservation is that these historical works were based upon lattice-gas models that are archaic as contributions to molecular science sometimes useful, but rarely an attempt to come to grips with molecular problems at a molecular resolution. Nevertheless, quasi-chemical and Bethe theories are probably the most effective approximate physical theories available for those lattice gas problems, and an important goal of our present discussion will be to discover corresponding theories of molecular science, as distinct from lattice gases. 

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